The McGee Group

This is a bit of a shaggy dog story, but I think it’s fun, and there’s a moral about the nature of mathematical research.

Act 1

Once I was interested in the McGee graph, nicely animated here by Mamouka Jibladze:

This is the unique (3,7)-cage, meaning a graph such that each vertex has 3 neighbors and the shortest cycle has length 7. Since it has a very symmetrical appearance, I hoped it would be connected to some interesting algebraic structures. But which?

I read on Wikipedia that the sym...